TY - JOUR
T1 - GAUSSIAN COMPLEX ZEROES ARE NOT ALWAYS NORMAL
T2 - LIMIT THEOREMS ON THE DISC
AU - Buckley, Jeremiah
AU - Nishry, Alon
N1 - Publisher Copyright:
© 2022 Mathematical Sciences Publishers.
PY - 2022
Y1 - 2022
N2 - We study the zeroes of a family of random holomorphic functions on the unit disc, distinguished by their invariance with respect to the hyperbolic geometry. Our main finding is a transition in the limiting behaviour of the number of zeroes in a large hyperbolic disc. We find a normal distribution if the covariance decays faster than a certain critical value. In contrast, in the regime of “long-range dependence” when the covariance decays slowly, the limiting distribution is skewed. For a closely related model we emphasise a link with Gaussian multiplicative chaos.
AB - We study the zeroes of a family of random holomorphic functions on the unit disc, distinguished by their invariance with respect to the hyperbolic geometry. Our main finding is a transition in the limiting behaviour of the number of zeroes in a large hyperbolic disc. We find a normal distribution if the covariance decays faster than a certain critical value. In contrast, in the regime of “long-range dependence” when the covariance decays slowly, the limiting distribution is skewed. For a closely related model we emphasise a link with Gaussian multiplicative chaos.
KW - Gaussian analytic functions
KW - Wiener chaos
KW - stationary point processes
UR - http://www.scopus.com/inward/record.url?scp=85153459328&partnerID=8YFLogxK
U2 - 10.2140/pmp.2022.3.675
DO - 10.2140/pmp.2022.3.675
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AN - SCOPUS:85153459328
SN - 2690-0998
VL - 3
SP - 675
EP - 706
JO - Probability and Mathematical Physics
JF - Probability and Mathematical Physics
IS - 3
ER -